In this article, we investigate a stochastic Galerkin method for the Maxwell equations with random inputs. The generalized Polynomial Chaos (gPC) expansion technique is used to obtain a deterministic system of the gPC expansion coefficients. The regularity of the solution with respect to the random is analyzed. On the basis of the regularity results, the optimal convergence rate of the stochastic Galerkin approach for Maxwell equations with random inputs is proved. Numerical examples are presented to support the theoretical analysis.
Lizheng CHENG
,
Bo WANG
,
Ziqing XIE
. A STOCHASTIC GALERKIN METHOD FOR MAXWELL EQUATIONS WITH UNCERTAINTY[J]. Acta mathematica scientia, Series B, 2020
, 40(4)
: 1091
-1104
.
DOI: 10.1007/s10473-020-0415-z
[1] Canuto C, Hussaini M Y, Quarteroni A, Zang T A. Spectral Methods Fundamentals in Single Domains. Berlin:Springer-Verlag, 2006
[2] Babuska I, Nobile F, Tempone R. A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J Numer Anal, 2007, 45:1005-1034
[3] Fishman G S. Monte Carlo, Concepts, Algorithms, and Applications. New York:Springer, 1996
[4] Fox B L. Strategies for Quasi-Monte Carlo. New York:Springer, 1999
[5] Ganapathysubramanian B and Zabaras N. Sparse grid collocation schemes for stochastic natural convection problems. J Comput Phys, 2007, 225:652-685
[6] Ghanem R G, Spanos P. Stochastic Finite Elements:A Spectral Approach. Dover, Revised, 2012
[7] Gottlieb D, Xiu D B. Galerkin method for wave equations with uncertain coefficients. Commun Comput Phys, 2008, 3:505-518
[8] Tatang M, McRae G. Direct Treatment of Uncertainty in Models of Reaction and Transport. MIT Tech Rep, 1994
[9] Liu W, Belytschko T, Mani A. Probabilistic finite elements for nonlinear structural dynamics. Comput Methods Appl Mech Eng, 1986, 56:61-81
[10] Shen J, Tang T, Wang L L. Spectral Methods:Algorithms, Analysis and Applications (Springer Series in Computational Mathematics). New York:Springer, 2011
[11] Tang T, Zhou T. Convergence analysis for stochastic collocation methods to scalar hyperbolic equations with a random wave speed. Commun Comput Phys, 2010, 8:226-248
[12] Xiu D B. Efficient collocational approach for parametric uncertainty analysis. Commun Comput Phys, 2007, 2:293-309
[13] Xiu D B. Numerical Methods for Stochastic Computations. New Jersey:Princeton University Press, 2010
[14] Wiener N. The homogeneous chaos. Amer J Math, 1938, 60:897-936
[15] Xiu D B. Fast numerical methods for stochastic computations:A review. Commun Comput Phys, 2009, 5:242-272
[16] Zhang G N, Gunzburger M. Error analysis of a stochastic collocation method for parabolic partial differential equations with random input data. SIAM J Numer Anal, 2012, 50:1922-1940
[17] Galindo D, Jantsch P, Webster C G, Zhang G N. Accelerating stochastic collocation methods for partial differential equations with random input data. SIAM/ASA J Uncertainty Quantification, 2015, 4:1111- 1137
[18] Chen L P, Zheng B, Lin G, Voulgarakis N. A two-level stochastic collocation method for semilinear elliptic equations with random cofficients. J Comput Appl Math, 2017, 315:195-207
[19] Hu J W, Jin S. A stochastic Galerkin method for the Boltzmann equation with un certainty. J Comput Phys, 2016, 315:150-168
[20] Chauviere C, Hesthaven J S, Lurati L. Computational modeling of uncertainty in time-domain electromagnetics. SIAM J Sci Comput, 2006, 28(2):751-775
[21] Benner P, Schneider J. Uncertainty quantification for Maxwell equations using stochastic collocation and model order reduction. Int J Uncertain Quantif, 2015, 5(3):195-208
[22] Horsin T, Stratis I, Yannacopoulos A. On the approximate controllability of the stochastic Maxwell equations. IMA J Math Control Inform, 2010, 27:103-118
[23] Liu H, Wu Q L, Cournede P H. Research on the effects of different sampling algorithm on Sobol sensitivity analysis. Acta Math Sci, 2018, 38A(2):372-384
[24] Duan L Q. The Gelfand approximations on generalized Besov class $B_{p,Q}\Omega$ in the deterministic and Monte Carlo settings. Acta Math Sci, 2012, 32A(1):148-160
[25] Hong J L, Ji L H, Zhang L Y. A stochastic multi-symplectic scheme for stochastic Maxwell equations with additive noise. J Comput Phys, 2014, 268:255-268
[26] Chauviere C, Hesthaven J S, Lurati L. Computational modeling of uncertainty in time-domain electromagnetics. SIAM J Sci Comput, 2006, 28(2):751-775
[27] Xiu D B, Karniadakis G E. The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J Sci Comput, 2002, 24(2):619-644
[28] Filbet F. On deterministic approximation of the Boltzmann equation in a bounded domain. Multiscale Model Simul, 2012, 10:792-817
[30] Pettersson M P, Iaccarino G, Nordstrom J. Polynomial Chaos Methods for Hyperbolic Partial Differential Equations. Springer, 2015
[30] Markowich P A, Ringhofer C, Schmeiser C. Semiconductor Equations. New York:Springer Verlag Wien, 1990
[31] Hu J, Jin S, Shu R. A stochastic Galerkin method for the Fokker-Planck-Landau equation with random uncertainties. Proc. 16th International Conference on Hyperbolic Problems, 2016, 237:1-19
[32] Jin S, Lu H. An asymptotic-preserving stochastic Galerkin method for the radiative heat transfer equations with random inputs and diffusive scalings. J Comput Phys, 2017, 334:182-206
[33] Jin S, Shu R. A stochastic asymptotic-preserving scheme for a kinetic-fluid model for disperse two-phase flows with uncertainty. J Comput Phys, 2017, 335:905-924
[34] Jichun Li, Zhiwei Fang, Guang Lin. Regularity analysis of metamaterial Maxwell's equations with random coefficients and initial conditions. Comput Methods Appl Mech Engrg, 2018, 335:24-51
